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Worksheet 1.5A, Function composition MATH 1410 (SOLUTIONS)
- Given the functions f and g, below, find the composition function f ◦ g. (The function (f ◦ g)(x)
is the same as f (g(x)).
(a) f (x) = x^2 ; g(x) =
x.
(b) f (x) =
x; g(x) = x^2.
(c) f (x) = x^2 − 1; g(x) = x + 2.
(d) f (x) = x + 2; g(x) = x^2 − 1.
(e) f (x) = x + 3 and g(x) = x 2 − 10
(f) f (x) = e x ; g(x) = x 2 .
(g) f (x) = x 2 ; g(x) = e x .
Solution.
(a) f (x) = x 2 ; g(x) =
x.
(f ◦ g)(x) = x
(b) f (x) =
x; g(x) = x 2 .
(f ◦ g)(x) = |x|
(c) f (x) = x^2 − 1; g(x) = x + 2.
(f ◦ g)(x) = (x + 2)^2 − 1 = x^2 + 4x + 3
(d) f (x) = x + 2; g(x) = x^2 − 1.
(f ◦ g)(x) = (x^2 − 1) + 2 = x^2 + 1
(e) f (x) = x + 3 and g(x) = x 2 − 10
(f ◦ g)(x) = (x 2 − 10) + 3 = x 2 − 7
(f) f (x) = e x ; g(x) = x 2 .
(f ◦ g)(x) = e x^2
(g) f (x) = x 2 ; g(x) = e x .
(f ◦ g)(x) = (e x ) 2 = e 2 x
- Given the functions f and g, below, find the composition functions f ◦ g and g ◦ f. (The function
(f ◦ g)(x) is the same as f (g(x)); (g ◦ f )(x) is the same as g(f (x)).)
(Please distinguish between your answer for f ◦ g and g ◦ f .)
(a) f (x) = x 2
(b) f (x) = x 3
3
(c) f (x) = x^2 + 9 and g(x) =
x.
(d) f (x) = x^2 + 6x + 9 and g(x) =
x.
(e) f (x) = x^2 + 5 and g(x) =
x − 5.
Solution.
(a) f (x) = x 2
(f ◦ g)(x) = f (g(x)) = f (
2
(g ◦ f )(x) = g(f (x)). But g(anything) =
3, so the answer is
(f ◦ g)(x) = 4 and (g ◦ f )(x) =
(b) f (x) = x^3 + 2 and g(x) = 3
(f ◦ g)(x) = f (g(x)) = f ( 3
3
5)^3 + 2 = 5 + 2 = 7.
(g ◦ f )(x) = g(f (x). But g(anything) = 3
- So (g ◦ f )(x) = 3
(f ◦ g)(x) = 7 and (g ◦ f )(x) =
(c) f (x) = x 2
x.
(f ◦ g)(x) = (
x) 2
(g ◦ f )(x) =
x^2 + 9.
(f ◦ g)(x) = x + 9 and (g ◦ f )(x) =
x^2 + 9.
(d) f (x) = x 2
x.
(f ◦ g)(x) = (
x) 2
x + 9 = x +
x + 9.
(g ◦ f )(x) =
x^2 + 6x + 9.
(f ◦ g)(x) = x + 6
x + 9 and (g ◦ f )(x) =
x^2 + 6x + 9.
(e) f (x) = x 2
x − 5.
(f ◦ g)(x) = (
x − 5) 2
(g ◦ f )(x) =
x^2 + 5 − 5 =
x^2 = |x|.
(f ◦ g)(x) = (
x − 5)^2 + 5 and (g ◦ f )(x) = |x|.
